Final Report Summary - GELATI (Geometry of exceptional Lie algebras à la Tits)
The magic square was constructed by Freudenthal and Tits starting from a pair (A, B) of composition algebras and forming a corresponding Lie algebra.
There exist interpretations (both split and non-split) of this table over arbitrary fields, as well as interpretations emphasizing the geometries related to these algebras and groups, i.e. the Tits buildings. In recent years the geometries appearing in the magic square have been studied quite exten- sively, mostly over the complex numbers, using algebraic geometry and representation theory. In 1984 Mazzocca and Melone formulated axioms involving conics which characterize finite quadratic Veronesean varieties over fields of odd order (the A2 case). In joint work with Van Maldeghem the applicant established that a more abstract version of these axioms turns out to be characteristic for the second row of the magic square over arbitrary fields.
The highlight of this research so far is the characterization of the split version of the second row over arbitrary fields as projective planes over split composition algebras [SVM4] 1. The varieties obtained are the analogues over arbitrary fields of the ones in the following theorem by Zak: Over an algebraically closed field of characteristic zero each Severi (secant defective) variety is projectively equivalent to either a quadratic Veronese variety in 5 dimensions, or a Segre variety in 8 dimensions, or a line Grassmannian variety in 14 dimensions, or the Cartan variety in 26 dimensions. In 1901 Severi already proved Zak’s theorem for surfaces. In 2014, in joint work with Krauss and Van Maldeghem the applicant obtained also a
characterization of the non-split version of the second row as the Veronesean representations of
Moufang planes over quadratic alternative division algebras. In particular this characterized
the Veronesean representation of the Moufang projective plane P(O) related to any Cayley-
Dickson division algebra O, which is the geometry of the real form E28 of the simple Lie group 6,2
of exceptional type E6.
There exist interpretations (both split and non-split) of this table over arbitrary fields, as well as interpretations emphasizing the geometries related to these algebras and groups, i.e. the Tits buildings. In recent years the geometries appearing in the magic square have been studied quite exten- sively, mostly over the complex numbers, using algebraic geometry and representation theory. In 1984 Mazzocca and Melone formulated axioms involving conics which characterize finite quadratic Veronesean varieties over fields of odd order (the A2 case). In joint work with Van Maldeghem the applicant established that a more abstract version of these axioms turns out to be characteristic for the second row of the magic square over arbitrary fields.
The highlight of this research so far is the characterization of the split version of the second row over arbitrary fields as projective planes over split composition algebras [SVM4] 1. The varieties obtained are the analogues over arbitrary fields of the ones in the following theorem by Zak: Over an algebraically closed field of characteristic zero each Severi (secant defective) variety is projectively equivalent to either a quadratic Veronese variety in 5 dimensions, or a Segre variety in 8 dimensions, or a line Grassmannian variety in 14 dimensions, or the Cartan variety in 26 dimensions. In 1901 Severi already proved Zak’s theorem for surfaces. In 2014, in joint work with Krauss and Van Maldeghem the applicant obtained also a
characterization of the non-split version of the second row as the Veronesean representations of
Moufang planes over quadratic alternative division algebras. In particular this characterized
the Veronesean representation of the Moufang projective plane P(O) related to any Cayley-
Dickson division algebra O, which is the geometry of the real form E28 of the simple Lie group 6,2
of exceptional type E6.